Mod 6 Lesson 1 Remediation Notes

1
Solving Exponential Equations
We know that 102 100 So, if we write 102
10x We know the right hand side of the
equation equals 100. Knowing that, we then can
say that x must equal 2.
This shows that if two powers have the same base
are equal (in this case the base is 10), then
the exponents must be equal.
If we wrote 101.23 10x2 We know at means
that 1.23 x 2. This is true because
the bases are the same, thus the exponents are
equal. Solving this equation, x - 0. 77
2
a)
c)
b)
What if the bases are not equal? See next slide
3
What if we have different bases?
17 10x2
Ideally, we would want both bases to be 10. We
need to rewrite 17 as a power of 10.
To rewrite 17 as a power of 10, use the log key
on your calculator. Since the log key on your
calculator is base 10, Entering log(17) is the
same as log10 17 which is equal to 1.23045.
Now we can write 101.23 10x2
1.23 x 2
-0.77 x and we have found x.
What if the base is not 10? See next slide
4
What if we do not have bases of 10?
16 2x
Ideally, we would want both bases to be 2. We
need to rewrite 16 as a power of 2.
To rewrite 16 as a power of 2 24
Now we can write 24 2x
4 x and we have found x.
5
Example 1 Solve the equation (Find the value
of x that makes this statement true. On the left
side of the equation the base is two. The number
16 can also be written as a power of 2.)
Since both sides of the equation have the same
base, the exponents must be equal.
6
Graphical Interpretation of Example 1 The
equation states that the value of
equals 8 for some value of x. Graph the two
functions and .
Find the intersection of these two points.
Algebraically we found that these two intersect
at (-2, 16). Look at the intersection
graphically.
7
c)
More Examples for Solving Exponential Functions
a)
3x
b)
3x
15x
8
e)
d)
9
g)
f)
10
Find x
Since both bases are the same, the exponents are
the same. You now have a quadratic to solve.
11
Find x
Find x
12
Inequalities

• Inequality Property of Exponents
• If b is a positive number other than 1
• bx gt by if and only if x gt y and bx lt
  by if and only if x lt y